AiAC|\Tm-l34g'
NATIONAL ADVISORY COMMITTEE
FOR AERONAUTICS
TECHNICAL MEMORANDUM 1348
ON THE REPRESENTATION OF THE STABILITY REGION IN
OSCILLATION PROBLEMS WITH THE AID OF THE
HURWrrZ DETERMINANTS
By E. Sponder
Translation of *Zur Darstellung des Stabilitatsgebietes bei
Schwingungsaufgaben mit Hilfe der Hurwitz-
Determinanten." Schweizer Archiv,
March 1950.
Washington
August 1952
^ b ^ fl ( 00 ;J'/ 'I
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
TECHNICAL MEMORANDUM 13^
ON THE REPRESENTATION OF THE STABILITY REGION IN
OSCILLATION PROBLEMS WITH THE AID OF THE
HURWITZ DETERMINANTS*
By E. Sponder
For oscillation phenomena which may also have an unstable course,
it is customary to represent the regions where stability or instability
prevails in a plane as functions of two parameters x and y.
In order to determine whether stability exists at any point of the
plane represented (thus, whether a disturbance of the oscillation phe-
nomenon considered is damped in its course) it must be investigated
whether all roots X. of the frequency (characteristic) equation
}J^ + a-j^X^"-"- + . . . + aj^ =
have a negative real part at this arbitrary point. The mathematical
condition for this is known to be that the n Hurwitz determinants D^
to Dn which are formed from the coefficients aj^ to a^^ and are
fimctions of the two parameters x and y mentioned before are all
positive for this point. Resulting from this criterion and completely
equivalent to it is the fact that the coefficients a-, to a^^ and only
a few certain Hurwitz determinants must be positive.
If, conversely, all roots \ of the frequency equation have a nega-
tive real part, all values D^ to T)-^ are positive. If one now visu-
alizes the point considered before (for which we assume stability to have
been established) as traveling in the representation plane of figure 1,
there vary with its parameters -x and y also the coefficients a-^
to a-^y, the real parts of the roots X, and finally the n Hurwitz-
determinants. If one arrives at a point where for instance a real root
*"Ziar Darstellung des Stabilitatsgebietes bei Schwingungsaufgaben
mit Hilfe der Hurwitz- Determinanten." Schweizer Archiv, March 1950,
pp. 93-96.
2 NACA TM 13^
disappears, a^j also disappears since |aj^ is the product of the values
of all roots; if one reaches, in contrast, a point where the real part
of a complex root becomes zero, it can be shown that then Dn_l becomes
zero. For every case, however, the product ^-dP-^-^ disappears which is
nothing else but the Hurwitz determinant of the nth degree
Sn^n-l = ^n
Thus the important theorem, the proof for which will be presented
later, is valid:
The limits of the (usually unique) stable region lie at
^n = a^Dn.! =0.
For graphical representation, it is therefore completely sufficient
to plot Dj^ = or more simply ^n ~ '^ ^^^ ■'^n-1 ~ "^ ^^ separate
limiting curves oX a region for which stability is known to prevail at
an arbitrary point, as illustrated by figure 1. If the limit Eji-l ~ ^
is exceeded, the course of the oscillation process is "dynamically"
unstable because a damping becomes negative; beyond the limit a^^ = 0,
one usually calls the' oscillation process "statically" unstable.
In particular, the following is valid for oscillation phenomena
which lead to frequency equations of the 4th degree (reference l).
The Hurwitz determinant of the 4th degree formed from the coeffi-
cients A to E of the frequency equation
AX^ + BX.3 + CX^ + BX + E =
reads
2 T,2^
D^ = ED3 = E(BCD - AD'^ - B'^E) = ER
with the expression in parentheses known as Routh's discriminant R; the
latter is therefore nothing else but the Hurwitz determinant of the
NACA TM 13^3
3rd degree. Since the coefficient A is usually +1, it is generally
valid as dynamic stability condition that the expression (BC - D)D - B%
turns out positive. The static stability is then guaranteed in addition
by E > so that the graphic representation of the region of figure 2
results.
Therewith, every requirement has been met; for it is impossible
that within the region denoted as stable a curve C = or D =
could r\in and perhaps still further reduce this region.
It is therefore completely superfluous to investigate further what
sign the other coefficients of the frequency equation have once the
boundaries for a region E = and R = have been fixed; however,
and this is important, it must be known that, at an arbitrary point of
this region, stability actually prevails. For this it is sufficient
to check for a point conveniently situated (for instance on an axis)
the signs of the remaining coefficients A to D of the frequency
equation which is equivalent with the fact that there all Hurwitz deter-
minants Di to 'Dl^. turn out positive. That this is important can, for
instance, be recognized from the fact that, in the region indicated on
the left in the plane of representation of figure 2, instability may
prevail in spite of E > and R > 0.
The advantage of these recognized facts given above does not result
in much simplification for frequency equations of the 4th, 5th, or even
6th degree; however, this advantage may save a great deal of unnecessary
calculations. The expressions for the Hurwitz determinants become more
and more cumbersome for higher degrees so that it will probably be a
very welcome facilitation for the outlining of the stability region if
the mere representation of the two curves a^ = and Dn-1 = will
be sufficient.
There now follows the proof of the theorem given before that the
limits of the (usually imique) stable region lie at D^ = ^n-'^n-l ~ ^'
1. For the present consideration, the existence of at least one
stable region is presupposed. Not even one need necessarily always
exist; on the other hand, s^eral stability regions, separated from
one another, may occur. One can easily make sure of this by considering
a frequency equation of the 2nd degree which pertains to an ordinary
oscillation
\
p
X, + a-.X + 32 =
MCA TM I3W
The coefficients a^^ and 82 are dependent on two parameters x
and y, corresponding to the two coordinate axes of the plane of repre-
sentation. For simplicity's sake, we shall assume that a2 everywhere
has positive values only; then the sign of a-^ alone is the criterion
for the stability, since a^ is a measvire of the damping of the oscil-
lation considered and this oscillation is stable only if a-, is posi-
tive. The limit of stability now depends entirely on the sign of the
analytical (or empirical) function of the coefficient a-, of the two
parameters x and y, or, in other words, what form the curves
=
have in the plane of representation which separate the positive from the
negative values. In this manner, one can easily understand that several
stability regions may exist but that just as well not even one need exist
anywhere in the entire plane of representation.
For frequency equations of higher degree, the probability decreases
that simultaneously in several regions all stability conditions found by
Hurwitz will be satisfied. Usually, one will deal with only one single
stability region which will then be considered more thoroughly.
2. Hiorwitz' criterion (reference 2) signifies that a prescribed
equation of the nth degree with real coefficients
iH
>n-l
ar,\ + anX,""-^ + . . . + a-„ =
(ao >0)
possesses roots exclusively with negative real parts only when the values
of the n-determinants
%
ai a3 a^
ao ag a^
an a-)
^2k-
-1
32k-
-2
32k-
■3
•
3k
(k = 1,2,. ..,n)
are all positive. Therein, one generally has to put a^ = when the
subscript x is negative or larger than n.
NACA TM 13^ 5
It shall now be proved that the Hiirwitz determinant of the (n - l)th
degree Dr,.! always disappears when the real part of a complex pair of
roots is zero.
Following the way Hurvitz piursued in deriving his criterion, one
finds the remarkable presupposition that no piorely imaginary roots must
exist. This, however, is precisely the condition whose influence on a
certain Hiirwitz -determinant is of special interest. However, for that
reason, one need not abandon this presupposition; it is sufficient to
interpret Ej^_i simply as a prescribed calculation rule which one applies,
without consideration of its derivation, to the coefficients of an equa-
tion of the nth degree.
It will now be expedient to consider (corresponding to the assump-
tion that the real part of a complex pair of roots is to be zero) the
equation of the nth degree as split into two factors: one is a quad-
ratic expression with the purely imaginary pair of roots and the other
is an expression of the (n - 2)th degree, thus
[x^ + a)(ao>^'^"^ + ajX^'^ + . . . + aj^_2) =
therein one may put the coefficient Sq = 1, without impairing the gen-
erality; the designation bq will, however, be retained. For all
remaining coefficients a2_ to an_2^ ^° additional restriction will be
prescribed other than that they are to be real and not all zero (which
would be trivial for this statement of the problem). Multiplication of
the two factors yields
aQ^^^ + a-LX,""-*- + (ag + aaQ)x^"^ + (a^ + aa^^)^'^"^ +
Aq Ai Ag Ao
The combined coefficients A one now visualizes as substituted in
the Hurwitz determinant Dn-i? "the general term of which has in the
NACA TM 13^
sth column and zth line the subscript x = 2s - z. Thus, one has there
Aps-z °-'^> expressed by the coefficients a, the term
32S-Z + aa2s-z-2
The value of the determinant Dn-l is now extended by multiplying
all terms of the first line by Bq and all terms of the second line
by a-]_. Fiirthermore, one maies use of the theorem that a determinant
does not change its value if one adds to the elements of one line the
elements of another line multiplied by an arbitrary number. This is done
by adding to the 'terms of the first line, which already have been multi-
plied by a„, the terms of the third, fifth, . . . line multiplied by
ap, a. , . . .j thus, one obtains as the general term of the sth col\amn
in the first line
3z-l(a2s-z + a32s_2_2)
Z— J_, J, p, • • •
Likewise, one adds to the terms of the second line already multiplied
by a-|^ the terms of the fourth, sixth, . . . line multiplied by a-D,
a^, . . .; one then finds as the general term of the sth column in the
second line
, , ^z-l(32s-z + aa2s-z-2)
z=2,4,6,...
In order to calculate these sums, it is sufficient to note that one
must generally put a^ = when the subscript x is negative; there
then results for a term of the first line
NACA TM 13^
Z_ az-l(a2s-z + 3a2s-z-2) = aoa2s-l + 32^23-3 + • • • +
^2s
_l^a3 + ags.pai + a(ao32s-3 + ^2^23-5 + • • • + 323-4^1)
and for a term of the second line
9z-l(a2s-z + 9^23-2-2) = 31323-2 + 33''23-if
z=2,i^,6,...
^2s-3''2 + ^23-1^0 ^ ^(^1^2s-4 ^ ^3^23-6 + • • • + 32g_3aQ)
One can 3ee that both 3ums are of equal magnitude which, however,
does not signify anything else but that the corresponding elements of
the first two lines are equal and that therefore the Hurv/itz-determinant
^n-1 extended by aQa-]_ identically disappears:
^O^lVl = °
From the derivation follows that one could have extended initially,
instead of the first and second line, two arbitrary other odd and even
lines by ag_-j_ and then have proceeded further in the same manner; one
then would have obtained quite generally
a a -^D , =
even odd n-1
8 MCA TM I3W
Under the obvious assumption that there always exists such a pair
of values of the coefficients a^ to an-2' ^^^ product of which does
not disappear, thus
Dj^_-L = 0, Q.E.D.
is valid.
3. From the derivation of this proof, one may f\irther conclude that
part of the curve Dji_i = may run entirely in the unstable region;
thus it does not appear there as the required stability limit. Since
such a possibility had been pointed out before, the motivation of this
noteworthy indication should be mentioned.
In splitting the prescribed equation of the nth degree into two
factors, no more detailed data on the coefficients bq to 82 °^
the expression of the (n - 2)th degree have been given and accordingly
none regarding the roots of the equation
,n-2 n-1
...)
either.
These roots may therefore have negative as well as positive real
parts in the neighborhood of points of the plane of representation for
which Dn-1 disappears; in other words, they may signify stability and
also instability. If there exists at least one root of the former
expression of the (n - 2)th degree with a positive real part, this fact
signifies that the curve E^.j = lies in an unstable region.
Figure 3 is to indicate how, for instance, three roots may vary
their position in the complex number plane if their point of reference
in the plane of representation shifts beyond the curve D^-l ~ ^' Thus
Dj^_j_ = is not necessarily always the boundary between two regions
where stability and instability prevail.
Neither does the sign of the coefficient a in the quadratic expres-
sion ( X,^ + a) play a role in the derivation of the proof. If a is
NACA TM 13^
positive, a purely imaginary pair of roots satisfies the equation
( A,2 + a ) =0; this case has been considered in particular. Hov/ever,
a could just as well be negative without altering the result of the
theorem. This signifies, however, that in the presence of two real
opposite-equal roots of the equation of the nth degree, the Hurwitz-
determinant of the (n - l)th degree also disappears.
As figure k shows, here also an unstable region lies to both sides
of the curve D^-i = so that again it does not appear as the sta-
bility limit.
Thus, these examples show that it is indispensable to make sure
whether really all stability conditions on one side of the curve
Dj^_-[_ = have been fulfilled; only then that curve forms together with
the curve aj^ = the boundary against the unstable region.
Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics
REFERENCES
1. Price, H. L. : The Lateral Stability of Aeroplanes. Aircr. Engng..
Vol. 15 (19^3), No. 173, 17^.
2. Hurwitz, A.: Ueber die Bedingungen, unter welchen eine Gleichung
nur Wurzeln mit negativen reellen Teilen besitzt. Mathematische
Annalen XLVI.
10
NACA TM I3I18
ii Dynamically unstable
Dn-l=0
o Point at which
stability prevails
an=0
Statically unstable
Figure 1.- Travel of the point after establishment of stability.
NACA TM 13i;8
11
R = '
E>0,R>0
Instability
possible _
Dynamically^
unstable
R=0
Stable
..^^rgr^?;^^^^^
E =
^^r*
Statically
unstable
Figure 2.- Graphic representation of the region E > 0.
12
NACA TM 13i+8
'n-1
NUnstableV^lv
-4t.
■-4r
NUnstobleOC^b:;
^ ^ ^ W \ s\
-^
^
Figure 3.- Change of position of the roots in the complex number plane.
1 =
W N NN \N N"'C
^
Nonstable W^_ ^^ ^'-^p
\;Unstable C^-t^ ©Xx
^ ^ \ \ \ \
Figure 4.- Unstable region for the curve
^n-l = 0-
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